Let $$f(x)=\sqrt{1+x}$$ Show that if $x \rightarrow 0$ a limit value does exist. Furtheremore, find the limit value and explain the choice of $\delta$ w.r.t $\epsilon$ when the definitions of a limit should be shown.
I need help with $\epsilon - \delta$-proofs and existence. I thought that showing a limit must exist was the whole point of $\epsilon - \delta$-proofs and that it would also specify the value by giving us a $\delta$ thats equal to some expression with $\epsilon$.
Would someone please walk me through how such a problem would be tackled? I.e how would you first show that a limit does indeed exist, formally. Then, find what it is and explain the choice of $\delta$ in the most simple way.
Thanks!